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# Comparison of Fractions, Decimals, and Percents

## Use <, > or = to compare fractions, decimals and percents.

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Comparison of Fractions, Decimals and Percents
Credit: Bonner Springs Library
Source: https://www.flickr.com/photos/bonnerlibrary/3739454987/

Steve and Nickie are at the library. They are reading the same book for a class. They make a bet to see who is the farthest along after 1 hour of reading. The winner decides what to eat for lunch. After an hour, Steve says he has read 35\begin{align*}\frac{3}{5}\end{align*} of the book. Nickie says she has read 65% of the book. Who will get to decide what to eat for lunch?

In this concept, you will learn to compare and order fractions, decimals, and percents.

### Comparing Fractions, Decimals, and Percents

You can compare fractions, decimals and percents using greater than, less than or equal to. Convert fractions, decimals, and percents to the same form to compare the values. If you are comparing a fraction and a percent, write both of them either as fractions or percents to figure out which is greater. Use this information to write a set of numbers or quantities in order from least to greatest or from greatest to least.

Compare 45% and 45\begin{align*}\frac{4}{5}\end{align*}.

First, write 45% and 45\begin{align*}\frac{4}{5}\end{align*} in the same form. Let’s change 45\begin{align*}\frac{4}{5}\end{align*} to a percent. Find the equivalent fraction of 45\begin{align*}\frac{4}{5}\end{align*} as a fraction out of 100. Multiply both numerator and denominator by 20.

4×205×20=80100\begin{align*}\frac{4 \times 20}{5 \times 20} = \frac{80}{100}\end{align*}

Next, convert the fraction to a percent.

80100=80%\begin{align*}\frac{80}{100} = 80\%\end{align*}

Now, we can compare 45% and 80%.

45%<80%\begin{align*}45\%<80\%\end{align*}

45% is less than 80%.

Let’s compare percents and decimals.

Compare 18% and 0.9.

First, write 18% and 0.9 in the same form. Let’s change 0.9 to a percent. Move the decimal point two places to the right and add the percent sign.

0.9=90%\begin{align*}0.9 =90\%\end{align*}

Now, compare 18% and 90%.

18%<90%\begin{align*}18\% < 90\%\end{align*}

18% is less than 90%.

Write 0.56, 34%, 910\begin{align*}\frac{9}{10}\end{align*}, and 12\begin{align*}\frac{1}{2}\end{align*} in order from least to greatest.

First, rewrite them in the same form. Let’s convert all of them to percents. 34% is already in percent form.

Convert 0.56 to a percent. Move the decimal point two places to the right and add the percent sign.

0.56=56%\begin{align*}0.56 = 56\%\end{align*}

Next, find the equivalent fractions of 910\begin{align*}\frac{9}{10}\end{align*} and 12\begin{align*}\frac{1}{2}\end{align*} as fractions over 100 and convert them to percents. Multiply the numerator and denominator of 910\begin{align*}\frac{9}{10}\end{align*} by 10. Multiply the numerator and denominator of 12\begin{align*}\frac{1}{2}\end{align*} by 50.

9×1010×101×502×50==90100=90%50100=50%\begin{align*}\begin{array}{rcl} \frac{9 \times 10}{10 \times 10} & = & \frac{90}{100} = 90\%\\ \frac{1 \times 50}{2 \times 50} & = & \frac{50}{100} = 50\% \end{array}\end{align*}

Then, compare the percents.

56%,34%,90%,50%\begin{align*}56\%, 34\%, 90\%, 50\%\end{align*}

Finally, write them in the order from least to greatest.

34%,50%,56%,90%  or34%,12,0.56,910\begin{align*}\begin{array}{rcl} && 34\%, 50\%, 56\%, 90\%\\ && \qquad \quad \ \ \text{or}\\ && 34\%, \frac{1}{2}, 0.56, \frac{9}{10} \end{array}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Steve and Nickie and their bet.

Steve has read 35\begin{align*}\frac{3}{5}\end{align*} of the book and Nickie has read 65% of the book. To find out who has read the farthest, compare the quantities.

First, rewrite 35\begin{align*}\frac{3}{5}\end{align*} as a percent. Find the equivalent fraction of 35\begin{align*}\frac{3}{5}\end{align*} as a fraction over 100 and convert to a percent. Multiply the numerator and denominator by 20.

3×205×20=60100=60%\begin{align*}\frac{3 \times 20}{5 \times 20} = \frac{60}{100} = 60\%\end{align*}

Next, compare the percents.

60%<65%\begin{align*}60\% < 65\%\end{align*}

60% is less than 35%. Nickie is farther along than Steve and will get to decide on lunch.

#### Example 2

Write the values in order from least to greatest.

83100,0.16,33%,45\begin{align*}\frac{83}{100} , 0.16, 33\%, \frac{4}{5}\end{align*}

First, rewrite then in the same form. Let’s write them as decimals. 0.16 is already in decimal form.

Convert 83100\begin{align*}\frac{83}{100}\end{align*} and 45\begin{align*}\frac{4}{5}\end{align*} to decimals. The fraction is 83 out of 100 or 83 hundredths. Convert 45\begin{align*}\frac{4}{5}\end{align*} to a decimal by dividing 4 by 5.

83100=0.835)4.0¯¯¯¯¯¯¯¯  0.8\begin{align*}\begin{array}{rcl} && \frac{83}{100} = 0.83\\ && \quad \overset{\ \ 0.8}{5\overline{) {4.0}}} \end{array}\end{align*}

Next, convert 33% to a decimal. Move the decimal point two places to the left and remove the percent sign.

33%=0.33\begin{align*}33\% = 0.33\end{align*}

Then, compare the decimals.

0.83,0.16,0.33,0.8\begin{align*}0.83, 0.16, 0.33, 0.8\end{align*}

Finally, write them in the order from least to greatest.

0.16,0.33,0.8,0.83or0.16,33%,45,83100\begin{align*}\begin{array}{rcl} && 0.16, 0.33, 0.8, 0.83\\ && \qquad \quad \text{or}\\ && 0.16, 33\%, \frac{4}{5}, \frac{83}{100} \end{array}\end{align*}

#### Example 3

Compare the values.

0.19 and 19%\begin{align*}0.19 \text{ and }1 9\%\end{align*}

First, rewrite them in the same form. Convert 19% to a decimal. Move the decimal two places to the left and remove the percent sign.

19%=0.19\begin{align*}19\% = 0.19\end{align*}

Next, compare the decimals.

0.19=0.19\begin{align*}0.19 = 0.19\end{align*}

0.19 and 19% are equal values.

#### Example 4

Compare the values.

25 and 45%\begin{align*}\frac{2}{5} \text{ and }45\%\end{align*}

First, rewrite them in the same form. Let’s convert both 25\begin{align*}\frac{2}{5}\end{align*} and 45% to decimals. Divide 2 by 5.

5)2.0¯¯¯¯¯¯¯¯   0.4\begin{align*}\overset{\ \ \ 0.4}{5 \overline{) {2.0}}}\end{align*}

Move the decimal place two places to the left and remove the percent sign.

45%=0.45\begin{align*}45\% = 0.45\end{align*}

Next, compare the decimals.

0.4<0.45\begin{align*}0.4 < 0.45\end{align*}

25\begin{align*}\frac{2}{5}\end{align*} is less than 45%.

#### Example 5

Compare the values.

56% and 21%\begin{align*}56\% \text{ and } 21\%\end{align*}

First, these values are in the same form. Compare the two percents.

56%>21%\begin{align*}56\% > 21\%\end{align*}

56% is greater than 21%.

### Review

Write the following values in order from least to greatest.

1. 16100,.27,53%,15\begin{align*}\frac{16}{100}, .27, 53\%, \frac{1}{5}\end{align*}
2. 99100,.30,68%,910\begin{align*}\frac{99}{100}, .30, 68\%, \frac{9}{10}\end{align*}
3. 18100,.99,87%,1020\begin{align*}\frac{18}{100}, .99, 87\%, \frac{10}{20}\end{align*}
4. 88100,.18,23%,15\begin{align*}\frac{88}{100}, .18, 23\%, \frac{1}{5}\end{align*}
5. 93100,.98,6%,12\begin{align*}\frac{93}{100}, .98,6\%, \frac{1}{2}\end{align*}
6. 77100,.37,93%,25\begin{align*}\frac{77}{100}, .37,93\%, \frac{2}{5}\end{align*}
7. \begin{align*}\frac{12}{100}, .76,13\%, \frac{1}{3}\end{align*}
8. \begin{align*}\frac{9}{100}, .2, 67\%, \frac{4}{5}\end{align*}
9. \begin{align*}\frac{88}{100}, .29, 35\%, \frac{2}{10}\end{align*}

Compare the following values using <, > or =.

1.  \begin{align*}\frac{6}{7}\end{align*} and 35%
2. \begin{align*}\frac{3}{4}\end{align*} and 75%
3. \begin{align*}\frac{1}{2}\end{align*} and 55%
4. \begin{align*}\frac{9}{10}\end{align*} and 25%
5. \begin{align*}\frac{1}{4}\end{align*} and 25%
6. \begin{align*}\frac{1}{5}\end{align*} and 15%

To see the Review answers, open this PDF file and look for section 8.16.

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Color Highlighted Text Notes

### Vocabulary Language: English

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).

Equivalent

Equivalent means equal in value or meaning.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

Order

Writing numbers in order commonly refers to writing them from least to greatest or greatest to least.

Percent

Percent means out of 100. It is a quantity written with a % sign.